05A05-RuleOfProduct.tex

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\pmtitle{rule of product}
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\pmsynonym{multiplication principle}{RuleOfProduct}
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\begin{document}
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If a process $A$ can have altogether $m$ different results and another process $B$ altogether $n$ different results, then the two processes can have altogether $mn$ different combined results.\, Putting it to set-theoretical form,
$$\mbox{card}(A\!\times\!B) \;=\; m\!\cdot\!n.$$

The \emph{rule of product} is true also for the combination of several processes:\, If the processes $A_i$ can have $n_i$ possible results ($i = 1,\,2,\,\ldots,\,k$), then their combined process has $n_1n_2\!\cdots\!n_k$ possible results.\, I.e.,
$$\mbox{card}(A_1\!\times\!A_2\!\times\ldots\times\!A_k) \;=\; n_1n_2\!\cdots\!n_k.$$\\


\textbf{Example.}\, Arranging $n$ elements, the first one may be chosen freely from all the $n$ elements, the second from the remaining $n\!-\!1$ elements, the third from the remaining $n\!-\!2$, and so on, the penultimate one from two elements and the last one from the only remaining element; thus by the rule of product, there are in all 
$$n(n\!-\!1)(n\!-\!2)\!\cdots\!2\!\cdot\!1 \;=\; n!$$ 
different arrangements, i.e. permutations, as the result.
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